Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless - - PowerPoint PPT Presentation

Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements

Seyedehsara Nayer, Praneeth Narayanamurthy, Namrata Vaswani

Iowa State University

Introduction Phase Retrieval (PR) Recover a length n signal x∗ from its phaseless linear projections yi := |ai, x∗|, i = 1, 2, . . . , m Without any structural assumptions, PR necessarily needs m ≥ n. To reduce sample complexity, can try to exploit structure Most existing work studies sparse PR – assumes x∗ is sparse.

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 2 / 8

Introduction Phase Retrieval (PR) Recover a length n signal x∗ from its phaseless linear projections yi := |ai, x∗|, i = 1, 2, . . . , m Without any structural assumptions, PR necessarily needs m ≥ n. To reduce sample complexity, can try to exploit structure Most existing work studies sparse PR – assumes x∗ is sparse. Another simple structure is low-rank. Two ways to use this:

1

assume x∗ can be rearranged as a low-rank matrix (not studied); or

2

assume a set of signals (or vectorized images) x∗

k , k = 1, 2, . . . , q,

together form a low-rank matrix The second is a more practical and commonly used model and we use this:

◮ first studied in our earlier work [Vaswani, Nayer, Eldar, Low-Rank Phase Retrieval, T-SP’17] Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 2 / 8

The Problem , [Nayer, Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)] Recover an n × q matrix of rank r X ∗ = [x∗

1 , x∗ 2 , . . . , x∗ k , . . . x∗ q ]

from a set of m phaseless linear projections of each of its q columns yik := |aik, x∗

k |, i = 1, . . . , m, k = 1, . . . , q.

Application: fast phaseless dynamic imaging, e.g., Fourier ptychographic imaging

• f live biological specimens

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 3 / 8

The Problem , [Nayer, Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)] Recover an n × q matrix of rank r X ∗ = [x∗

1 , x∗ 2 , . . . , x∗ k , . . . x∗ q ]

from a set of m phaseless linear projections of each of its q columns yik := |aik, x∗

k |, i = 1, . . . , m, k = 1, . . . , q.

Application: fast phaseless dynamic imaging, e.g., Fourier ptychographic imaging

• f live biological specimens

Even the linear version of this problem is different from both

◮ LR matrix sensing: recover X ∗ from yi = Ai, X ∗ with Ai’s dense ⋆ global measurements (yi depends on entire X ∗) ◮ LR matrix completion: recover X ∗ from a subset of its entries ⋆ completely local measurements ⋆ need rows & cols to be “dense” to allow for correct “interpolation”

Our problem - non-global measurements of X ∗, but global for each column

◮ only need denseness of rows (incoherence of right singular vectors) Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 3 / 8

Main Result for AltMinLowRaP[Nayer, Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)] Recover an n × q rank-r matrix X ∗ from yik = |aik, x∗

k |, i ∈ [1, m], k ∈ [1, q].

AltMinLowRaP algo: careful spectral init followed by alternating minimization.

Theorem (Guarantee for AltMinLowRaP)

Assume µ-incoherence of right singular vectors of X ∗. Set T := C log(1/ǫ). Assume that, for each new update step, we use a new (independent) set of mq measurements with m satisfying mq ≥ Cκ6µ2 nr 4 and m ≥ C max(r, log q, log n). Then, w.p. at least 1 − 10n−10, dist(ˆ xT

k , x∗ k ) ≤ ǫx∗ k , k = 1, 2, . . . , q

Also, the error decays geometrically with t. Sample complexity: C · nr 4 log(1/ǫ) (treating κ, µ as constants). Time complexity: C · mqnr log2(1/ǫ).

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 4 / 8

Discussion [Nayer, Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)] Recover a rank-r n × q matrix X ∗ from yik = |aik, x∗

k |, i ∈ [1, m], k ∈ [1, q].

Treating κ, µ as constants, our sample complexity is mtotq ≥ C nr 4 log(1/ǫ) Number of unknowns in X ∗ is (q + n)r ≈ 2nr

◮ sample complexity is r 3 times the optimal value (nr)

No existing guarantees for our problem or even its linear version:

◮ closest LR recovery problem with non-global measurements is LR

Matrix Completion (LRMC)

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 5 / 8

Discussion [Nayer, Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)] Recover a rank-r n × q matrix X ∗ from yik = |aik, x∗

k |, i ∈ [1, m], k ∈ [1, q].

Treating κ, µ as constants, our sample complexity is mtotq ≥ C nr 4 log(1/ǫ) Number of unknowns in X ∗ is (q + n)r ≈ 2nr

◮ sample complexity is r 3 times the optimal value (nr)

No existing guarantees for our problem or even its linear version:

◮ closest LR recovery problem with non-global measurements is LR

Matrix Completion (LRMC) Sample complexity of non-convex LRMC solutions is also sub-optimal

◮ AltMinComplete needs C nr 4.5 log(1/ǫ) samples ◮ Best LRMC solution (proj-GD) needs C nr 2 log2 n samples

Comparison with standard (unstructured) PR

◮ Standad PR sample complexity is nq: much larger when r 4 ≪ q Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 5 / 8

Key idea of the algorithm: Alt-Min for Phaseless Low Rank Recovery [Nayer,

Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)]

Alternating minimization relies on the following key idea:

1

Let X ∗ = U∗B∗. Thus x∗

k = U∗b∗ k and so yik := |aik, x∗ k | = |U∗′aik, b∗ k|

2

If U∗ is known, recovering b∗

k is an (easy) r-dimensional standard PR

problem

⋆ needs only m ≥ r measurements. 3

Given an estimate of U∗ and of b∗

k, we can get an estimate of phase of

aik, x∗

k . Updating U∗ is then a Least Squares problem

⋆ can show that for this step mq of order nr 4 suffices. Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 6 / 8

Key idea of the algorithm: Alt-Min for Phaseless Low Rank Recovery [Nayer,

Narayanamurthy, Vaswani, Phaseless PCA, ICML 2019 (this work)]

Alternating minimization relies on the following key idea:

1

Let X ∗ = U∗B∗. Thus x∗

k = U∗b∗ k and so yik := |aik, x∗ k | = |U∗′aik, b∗ k|

2

If U∗ is known, recovering b∗

k is an (easy) r-dimensional standard PR

problem

⋆ needs only m ≥ r measurements. 3

Given an estimate of U∗ and of b∗

k, we can get an estimate of phase of

aik, x∗

k . Updating U∗ is then a Least Squares problem

⋆ can show that for this step mq of order nr 4 suffices.

Spectral init: compute ˆ Uinit as top r eigenvectors of YU = 1 mq

q

• k=1

m

• i=1

y 2

ikaikaik′1 y 2

ik≤ 9 mq

• ik y 2

ik

• Nayer, Narayanamurthy, Vaswani (Iowa State Univ)

Phaseless PCA 6 / 8

AltMin-LowRaP: Alt-Min for Phaseless Low Rank Recovery [Nayer, Narayanamurthy, Vaswani,

Phaseless PCA, ICML 2019 (this work)]

1: ˆ

r ← largest index j for which λj(YU) − λn(YU) ≥ ω

2: U ← top ˆ

r singular vectors of YU :=

1 mq

• i,k:y 2

ik≤ 9 mq

• ik y 2

ik y 2

ikaikaik ′

3: for t = 0 : T do 4:

ˆ bk ← RWF({yk, U′aik}, i = 1, 2, . . . , m) for each k = 1, 2, · · · , q

5:

ˆ X t ← U ˆ B where ˆ B = [ˆ b1, ˆ b2, . . . ˆ bq]

6:

QR decomposition: ˆ B

QR

= RBB

7:

ˆ cik ← phase(aik, ˆ xik), i = 1, 2, . . . , m, k = 1, 2, · · · , q

8:

ˆ U ← arg min ˜

U

q

k=1

m

i=1(ˆ

cikyik − aik ′ ˜ Ubk)2

9:

QR decomp: ˆ U

QR

= URU

10: end for

RWF: one of two (provably) best standard PR methods

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 7 / 8

Selected Real Video Results – I

(a) Original (b) AltMinLowRaP (c) RWF

Figure 1: Recovering a real video of a moving mouse (approx low-rank) from simulated m = 5n coded diffraction pattern (CDP) measurements. Showing frames 20, 60, 78.

Nayer, Narayanamurthy, Vaswani (Iowa State Univ) Phaseless PCA 8 / 8